definition. algebra [ca-0016]

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Algebra

L∃∀N

Let \(R\) be a commutative ring. An algebra \(A\) over \(R\) is a pair \((A, \bullet )\), satisfying:

1. \(A\) is a ring under \(*\).
2. there exists a ring homomorphism from \(R\) to \(A\), denoted \(\mathit {1} : R \to _{+*} A\).
3. \(\bullet : R \to M \to M\) is a scalar multiplication
4. for every \(r \in R\), \(x \in A\), we have
   1. \(r * x = x * r\) (commutativity between \(R\) and \(A\))
   2. \(r \bullet x = r * x\) (definition of scalar multiplication)
   where we omitted that the ring homomorphism \(\mathit {1}\) is applied to \(r\) before each multiplication.